This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by LeJan and Sznitman [Probab. Theory Related Fields 109 (1997) 343–366] and extended by Bhattacharya et al. [Trans. Amer. Math. Soc. 355 (2003) 5003–5040; IMA Vol. Math. Appl., vol. 140, 2004, in press]. In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration.

A number of papers have examined various aspects of “random random walks” on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this work. This article also provides a number of exercises. Some exercises involve straightforward computations; others involve proving details in proofs or extending results proved in the article. This article also describes some problems for further study.

This is a survey paper on Poisson approximation using Stein’s method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector’s problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered.

This is an update of, and a supplement to, a 1986 survey paper by the author on basic properties of strong mixing conditions.

This survey paper is based on mini-courses given in July 2003 at the University of St-Andrews, and at the ICMS in Edinburgh. Its goal is to give a self-contained and heuristic overview of the recent (i.e. pre-2003) results concerning conformal restriction properties for random planar curves.

This text surveys properties and applications of the exponential functional $\int^t_0$exp$(-\xi_s)ds$ of real-valued Lévy processes $\xi = (\xi_t, t \geq 0)$.

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.

We discuss several connections between discrete and continuous random trees. In the discrete setting, we focus on Galton-Watson trees under various conditionings. In particular, we present a simple approach to Aldous’ theorem giving the convergence in distribution of the contour process of conditioned Galton-Watson trees towards the normalized Brownian excursion. We also briefly discuss applications to combinatorial trees. In the continuous setting, we use the formalism of real trees, which yields an elegant formulation of the convergence of rescaled discrete trees towards continuous objects. We explain the coding of real trees by functions, which is a continuous version of the well-known coding of discrete trees by Dyck paths. We pay special attention to random real trees coded by Brownian excursions, and in a particular we provide a simple derivation of the marginal distributions of the CRT. The last section is an introduction to the theory of the Brownian snake, which combines the genealogical structure of random real trees with independent spatial motions. We introduce exit measures for the Brownian snake and we present some applications to a class of semilinear partial differential equations.

This paper is the first part of our survey on various results about the distribution of exponential type Brownian functionals defined as an integral over time of geometric Brownian motion. Several related topics are also mentioned.

This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.

We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others.

Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther and also comprise the zeros of the Riemann zeta function. The existing proofs require a substantial technical machinery and heavy tools from various parts of mathematics, in particular complex analysis, combinatorics and variational analysis. Particularly in the last decade, a number of fine results have been achieved, but it is obvious that a comprehensive and thorough understanding of the matter is still lacking. Hence, it seems an appropriate time to provide a surveying text on this research area.

In the present text, we introduce various models, explain the questions and problems, and point out the relations between the models. Furthermore, we concisely outline some elements of the proofs of some of the most important results. This text is aimed at non-experts with strong background in probability who want to achieve a quick survey over the field.

This paper contains sharp estimates about the distribution of multiple random integrals of functions of several variables with respect to a normalized empirical measure, about the distribution of U-statistics and multiple Wiener–Itô integrals with respect to a white noise. It also contains good estimates about the supremum of appropriate classes of such integrals or U-statistics. The proof of most results is omitted, I have concentrated on the explanation of their content and the picture behind them. I also tried to explain the reason for the investigation of such questions. My goal was to yield such a presentation of the results which a non-expert also can understand, and not only on a formal level.

This paper is a survey on some recent aspects and developments in stochastic control. We discuss the two main historical approaches, Bellman’s optimality principle and Pontryagin’s maximum principle, and their modern exposition with viscosity solutions and backward stochastic differential equations. Some original proofs are presented in a unifying context including degenerate singular control problems. We emphasize key results on characterization of optimal control for diffusion processes, with a view towards applications. Some examples in finance are detailed with their explicit solutions. We also discuss numerical issues and open questions.